Let If and are respectively the number of points at which the curves and intersect the axis, then the value of is

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Solution

Q.
Let $f (x) = 2^{x} - x^{2}, x \in R .$ If $m$ and $n$ are respectively the number of points at which the curves $y = f (x)$ and $y = f' (x)$ intersect the $x -$ axis, then the value of $m + n$ is _________. [2024]

Ans.
(5)

We have, $f (x) = 2^{x} - x^{2}$

By graph, since $f (x)$ intersects the $x$ -axis at 3 points. So, number of solutions of $f (x) = 3$

$∴$ $m = 3$

Also, $f' (x) = 2^{x} \log 2 - 2 x$

By graph, since $f' (x)$ intersects $x$ -axis at 2 points. So, number of solutions of $f' (x) = 2$

$∴$ $n = 2$

Thus, $m + n = 3 + 2 = 5$

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